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標題: S-平衡構型在R^n空間中的推廣
Generalized S-Balanced Configuration in R^n
作者: 沈妤庭
Yu-Ting Shen
關鍵字: 中心構型;牛頓N體運動問題;S-平衡構型;Central configuration;Newtonian N-body problem;S-balanced configuration
引用: [1] Albouy, A., & Kaloshin, V. Finiteness of central configurations of five bodies in the plane, Annals of Math., 176(2012), 535-588. [2] Hampton, M., & Jensen, A. Finiteness of spatial central configurations in the 5 body problem, Celest. Mech. Dyn. Astr. 109(2011), 321-332. [3] Hampton, M., & Moeckel, R. Finiteness of relative equilibria of the four-body problem, Inv. Math., 163(2006), 289-312. [4] Moeckel, R. (2014).LECTURES ON CENTRAL CONFIGURATIONS [5] Smale, S. Mathematical problems for the next century, Mathematical Intelligencer, 20(1998), 7-15. [6] Weisstein, E. W. 'Stanley's Theorem.' From MathWorld–A Wolfram Web Resource. publsihed online (2018), from [7] Wilf, H. S. (2000).Lectures on Integer Partitions, University of Pennsylvania, Philadelphia, PA, U.S. state.
研究牛頓N體問題中,我們可以去找出星體的初始位置,並給定特定初始速度,即可形成一些特殊解。在這篇論文中,我們將介紹中心構型以及它的一些推廣,中心構型是一組代數方程式的零根,其中質量為其方程組的參數,中心構型即是一些特殊解的初始位置,此構型的一個推廣為S-平衡構型,其中S是一個d × d之對角矩陣,而中心構型是當S是一個d × d之單位矩陣的情況。我們研究二體問題的S-平衡構型,特別是d = 2, 3, 4的情況,並對一般的d來推論結果。研究二體的問題目的在於對共線的部分解來加以了解。另外,我們使用Mathematica軟體來估計一些三體問題的S-平衡構型,我們討論當S =diag(1,2)時的情況,與平面三體中心構型不同,我們發現S-平衡構型的個數不是唯一,有可能為2或3。

To study Newtonian N-body problem, one can find the initial position of the 'n' bodies and provide special initial velocities to form special solutions. In this thesis, we will introduce Central configurations and their generalizations. It is a system of algebraic equations, with masses as parameters. Its common zero gives the initial positions of the special solutions.
One generalization of such configurations is called the S-balanced configurations for a given d × d diagonal matrix S. For central configurations, S is the d × d iden- tity matrix. In this theses, we study two-body S-balanced configurations for general potential to see how collinear central configurations as parts of the S-balanced configurations in R looks like. In particular, we focus on d = 2, 3, 4 and also conclude for general d. Also, we will compute numerically some three-body S-balanced configurations in R^2 with Mathematica. Different from the unique central configuration in R^2 for three bodies, we found that there can also be 2 or 3 planar S-balanced configurations. Here we consider cases S =diag(1,2).
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